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QUESTION: I do not know what to do at all with this number. Any help would be greatly appreciated.

Let vector x = <x_1, x_2> , vector y = <y_1, y_2, y_3> and vector z = <z_1, z_2>.

Say that the systems

x_1 = 2y_1 + 3y_2 - 5y_3

x_2 = -3y_1 - 4y_2 + 2y_3

and

y_1 = 2z_1 - 3z_2

y_2 = 3z_1 - z_2

y_3 = z_1 + z_2

(a) Rewrite these linear systems as matrix equations involving vector x, vector y, vector z.

(b) Use the matrix product to write vector x in terms of vector z.

ANSWER: a) Its easy to write down the matrix eqns, just pick out the coefficients of the eqns.

2y1+3y2-5y3 = x1

-3y1-4y2+2y3 = x2

or Ay = x

A = (2 3 -5)

(-3 -4 2)

y = (y1, y2, y3)^T

x = (x1,x2)^T

Also Bz = y

B = (2 -3)

(3 -1)

(1 1)

z = (z1,z2)^T.

b) Since Bz = y and Ay = x, we have

A(Bz) = Cz = x and all you have do do is multiply the 2x3 matrix A by the 3x2 matrix B in the usual way to get a 2x2 matrix C.

---------- FOLLOW-UP ----------

QUESTION: i was wondering if what i am doing for part a) is correct

x_1 - 2y_1 - 3y_2 + 5y_3 =0

x_2 +3y_1 + 4y_2 - 2y_3 =0

0x_1+0x_2+y_1 - 2z_1 + 3z_2=0

0x_1+0x_2+0y_1+y_2 - 3z_1 + z_2 =0

0x_1+0x_2+0y_1+0y_2+y_3 - z_1 - z_2 =0

When put into a matrix:

[x_1]

[1 0 -2 -3 5 0 0] [x_2] = [0]

[0 1 3 4 -2 0 0] [y_1] = [0]

[0 0 1 0 0 -2 3] [y_2] = [0]

[0 0 0 1 0 -3 1] [y_3] = [0]

[0 0 0 0 1 -1 -1] [z_1] = [0]

[z_2]

Thanks for any help

I'm not sure what you are doing. I gave the matrix and vector expressions for part a) in my response. Do you not understand this part? Are you really trying to do part b)? If so, then take the matrices A (2x3) and B(3x2) in my answer and calculate the matrix product C = AB, which is a 2x2 matrix and you can then write Cz = x since z and x are 2-component vectors.

If you do not know how to multiply matrices then look in your textbook, lecture notes, website or whatever to find out. This is a very basic calculation and you should learn it.

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